Strict and weak stationarity for integer-indexed time series

Let $X=(X_t{)}_{t\in \mathbb{Z}}$ be a real-valued stochastic process on one probability space, so each $X_t$ is a real random variable and each finite tuple has its pushforward law (Stochastic processes and their finite-dimensional distributions, Random elements and real random variables, Law or distribution of a random element). We call $X$ a time series here because its index set is $\mathbb{Z}$.

The process is strictly stationary when for every integer $r\ge 1$, every $t_1,\dots ,t_r\in \mathbb{Z}$, and every shift $m\in \mathbb{Z}$, $\mathcal{L}(X_{t_1+m},\dots ,X_{t_r+m})=\mathcal{L}(X_{t_1},\dots ,X_{t_r})\mathrm{.}$ This condition concerns the full finite-dimensional laws and imposes no moment condition by itself.

The process is weakly stationary when $E[\mathrm{\mid }{X}_t{\mathrm{\mid }}^2]<\mathrm{\infty }$ for every $t\in \mathbb{Z}$, its mean $\mu :=E[X_t]$ is independent of $t$, and for every $h\in \mathbb{Z}$ the covariance $\gamma (h):=\mathrm{Cov}(X_{t+h},X_t)$ is independent of $t$. Expectation, variance, and covariance have the meanings in Expectation of a nonnegative or integrable random variable and Moments, variance, and covariance on a probability space. Strict stationarity alone therefore does not supply the moment hypotheses needed for covariance or sample-mean claims.